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I have the following Model and would like to know how to acquire the OLS-estimates when the model is expressed like this? I have tried to look it up, but no one is explicit about how to acquire the OLS estimates when you have a linear coefficient and the rest a non-linear expression. I know that if there was no 1 in the denominator, I would be able to take the logarithm of y, which would make the model linear in parameters, but the 1 is kinda irritating me at this point.

If I take logs, I get the following: $$ \log(y) = -\log(1+\beta_0 + \beta_1 x + u) $$ and this does not help me a lot because the the model is not linear in parameters.

Any help would be much appreciated

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    $\begingroup$ Welcome! How about $1/y = 1+\exp(\beta_0+\beta_1+u)$ so $\log(1/y-1) = \beta_0+\beta_1 x+u$? It is required that $0<y<1$. $\endgroup$
    – chan1142
    Jun 11, 2023 at 1:01
  • $\begingroup$ You can try least-squares. $\endgroup$ Jun 11, 2023 at 7:08

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You can't estimate a model like this by OLS because it is not linear in parameters.

Are you willing to model this as shown below? $$y_i =\frac{1}{1+e^{\beta_0+\beta_1x_i}}+u_i $$

If so you could use nonlinear least squares. This can be implemented in Stata with the nl command.

If you are not able to change the modeling assumption, then I think you'd have to do method of moments. One moment would presumably be $E[u]=0$. The other moment would presumably be $E[xu]=0$. You'd have to code this yourself in python or matlab I think. You'd solve for the $\hat{\beta_0}$ and $\hat{\beta_1}$ such that the sample average of $\hat{u}$ is 0 and the sample average of $x_i\hat{u_i}$ is 0.

EDIT: You can do the method of moments in Stata. By solving for $u$, it is the case that $$u_i =\ln\left(\frac{1-y_i}{y_i} -\beta_0-\beta_1x_i\right)$$

Thus, the Stata code to do method of moments based on $E[u]=0$ and $E[xu]=0$ is

gmm (ln((1-y)/y) -{b0} -{b1}*x ), instruments(x)

You'll just need to change the names of $x$ and $y$ based on what they are called in your data.

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